Facing a simple, yet frustrating formula like this
and the task to solve it for x left me googling around for hours until I found salvation in Wolfram Alpha, Wikipedia, and a nice blog post with R-syntax to solve a similar equation.
Using the results from Wolfram Alpha I was able to find the solution with the ‘gsl’ library
# create some example data
dat <- data.frame(a = 0.109861, x = 10)
# a is set so that b is roughly 30.
# Lazy as I am I used Excel and its solver ability to find numbers
# to check if b is close to 30. Using the initial formula
dat$b <- dat$x * exp(dat$a * dat$x)
# solve for x2 and see if x and x2 are similar and close to 10
dat$x2 <- lambert_W0(dat$a * dat$b)/dat$a
# a x b2 x2
#1 0.109861 10 29.99993 10.00001
Sometimes life can be so easy (after a long time searching for the right results….).
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